1Universidad Nacional de Colombia, Bogotá, Colombia. Email: tapiens@gmail.com
]]> Abstract
We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution of the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.
Key words: Curvature tensors, mean curvature flow.
Se obtienen las ecuaciones de evolución para el tensor de Riemann, el tensor de Ricci y el escalar de curvatura inducidas por el flujo de curvatura media. La evolución de la curvatura escalar es similar al flujo de Ricci, sin embargo, la curvatura negativa, en vez de la positiva, es favorecida. Nuestros resultados son válidos en cualquier dimensión.
Palabras clave: Tensores de curvatura, flujo de curvatura media.
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References
[1] C. Burstin, `Ein beitrag zum problem der einbettung der riemannschen raume in euklidischen raumen´, Rec. Math. Moscou (Math. Sbornik) 38, (1931), 74-85. [ Links ]
[2] E. Cartan, `Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien´, Ann. Soc. Polon. Math. 6, (1927), 1-7. [ Links ]
[3] C. J. Clarke, `On the isometric global embedding of pseudo-riemannian manifolds´, Proc. Roy. Soc. London A 314, (1970), 417-428. [ Links ]
[4] K. Ecker, `Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in minkowski space´, J. Diff. Geom. 46, (1997), 481-498. [ Links ]
[5] J. Eells and J. H. Sampson, `Harmonic mappings of riemannian manifolds´, Am. J. Math. 86, (1964), 109-160. [ Links ]
[6] L. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, 1926. [ Links ]
[7] A. Friedman, `Local isometric embedding of riemannian manifolds with indefinite metric´, J. Math. Mech. 10, (1961), 625-649. [ Links ]
[8] R. E. Greene, `Isometric embedding of riemannian and pseudo-riemannian manifolds´, Memoirs Am. Math. Soc. 97, (1970), 1-63. [ Links ]
[9] M. Gunther, `On the perturbation problem associated to isometric embeddings of riemannian manifolds´, Ann. Global Anal. Geom. 7, (1989), 69-77. [ Links ]
[10] M. Gunther, Isometric embedding of riemannian manifolds, `Proceedings of the International Congress of Mathematicians´, (1990), Kyoto. [ Links ]
[11] R. Hamilton, `Three-manifolds with positive Ricci curvature´, J. Diff. Geom. 17, (1982), 255-306. [ Links ]
[12] G. Huisken, `Flow by mean curvature of convex surfaces into spheres´, J. Diff. Geom. 20, (1984), 237-266. [ Links ]
[13] M. Janet, `Sur la possibilité du plonger un espace riemannien donné dans un espace euclidien´, Ann. Soc. Pol. Math. 5, (1926), 38-43. [ Links ]
[14] J. Nash, `The imbedding problem for riemannian manifolds´, Ann. Math. 63, (1956), 20-63. [ Links ]
[15] G. Perelman, `The entropy formula for the ricci flow and its geometric applications´, arXiv:math.DG/0211159, 2002. [ Links ]
[16] G. Perelman, `Finite extinction time for the solutions to the Ricci flow on certain three-manifolds´, arXiv:math.DG/0307245, 2003a. [ Links ]
[17] G. Perelman, `Ricci flow with surgery on three-manifolds´, arXiv:math.DG/0303109, 2003b. [ Links ]
[18] J. A. Schouten, Ricci Calculus, Springer, Berlin, 1954. [ Links ]
[19] V. Tapia, `The geometrical meaning of the harmonic gauge for minimal surfaces´, Nuovo Cimento B 103, (1989), 441-445. [ Links ]
[20] W. P. Thurston, `Three dimensional manifolds, kleinian groups and hyperbolic geometry´, Bull. Am. Math. Soc. 6, (1982), 357-381. [ Links ]
[21] M. T. Wang, `Long-time existence and convergence of graphic mean curvature flow in arbitrary dimension´, Inv. Math. 148, (2002), 525-543. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCMv43n2a06,
AUTHOR = {Tapia, Víctor},
TITLE = {{Evolution of curvature tensors under mean curvature flow}},
JOURNAL = {Revista Colombiana de Matemáticas},
YEAR = {2009},
volume = {43},
number = {2},
pages = {175-185}
}